Simplify: [tex]$\frac{x^2+4 x-21}{x^2+6 x-7} \div \frac{5 x-15}{x-x^2}$[/tex]
Answer (1)
Rewrite the division as multiplication by the reciprocal.
Factorize the quadratic and linear expressions.
Cancel out the common factors.
Simplify the expression to get the final answer: − 5 x .
Explanation
Understanding the Problem We are asked to simplify the expression x 2 + 6 x − 7 x 2 + 4 x − 21 ÷ x − x 2 5 x − 15 . This involves dividing two rational expressions, which we can simplify by factoring and cancelling common terms.
Rewrite as Multiplication First, we rewrite the division as multiplication by the reciprocal: x 2 + 6 x − 7 x 2 + 4 x − 21 × 5 x − 15 x − x 2
Factorize Quadratics Next, we factorize the quadratic expressions: x 2 + 4 x − 21 = ( x + 7 ) ( x − 3 ) x 2 + 6 x − 7 = ( x + 7 ) ( x − 1 )
Factorize Linears Now, we factorize the linear expressions: x − x 2 = x ( 1 − x ) 5 x − 15 = 5 ( x − 3 )
Substitute Factors Substitute the factorized expressions into the expression: ( x + 7 ) ( x − 1 ) ( x + 7 ) ( x − 3 ) × 5 ( x − 3 ) x ( 1 − x )
Cancel Common Factors Cancel out the common factors ( x + 7 ) and ( x − 3 ) :
( x − 1 ) 1 × 5 x ( 1 − x )
Rewrite (1-x) Rewrite ( 1 − x ) as − ( x − 1 ) :
( x − 1 ) 1 × 5 − x ( x − 1 )
Final Simplification Simplify further by cancelling ( x − 1 ) :
1 1 × 5 − x = − 5 x
Final Answer Therefore, the simplified expression is − 5 x .
Examples
Rational expressions are used in many areas of science and engineering. For example, in physics, they can be used to describe the relationship between voltage, current, and resistance in an electrical circuit. Simplifying these expressions can help engineers to design more efficient circuits. In economics, rational expressions can be used to model cost and revenue functions. Simplifying these expressions can help businesses to make better decisions about pricing and production.
